Simplify the following expression and state the condition under which the simplification is valid: $z = \dfrac{t^2 - 6t}{t^2 - 5t - 6}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{t^2 - 6t}{t^2 - 5t - 6} = \dfrac{(t)(t - 6)}{(t + 1)(t - 6)} $ Notice that the term $(t - 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(t - 6)$ gives: $z = \dfrac{t}{t + 1}$ Since we divided by $(t - 6)$, $t \neq 6$. $z = \dfrac{t}{t + 1}; \space t \neq 6$